For a square matrix A, if
AB = BA = I
Then, B is the inverse of A
i.e. B = A ^{ −1 }
We will find inverse of a matrix by
Note: Since AB = BA = I
We can say B is the inverse of A.
i.e. B = A ^{ −1 }
We can also say,
A is the inverse of B
i.e. A = B ^{ −1 }
Thus, for inverse
We can write
AA ^{ −1 } = A ^{ −1 } A = I
Where I is identity matrix of the same order as A
Let’s look at same properties of Inverse.
Properties of Inverse

For a matrix A,
A ^{ −1 } is unique, i.e., there is only one inverse of a matrix  (A ^{ −1 } ) ^{ −1 } = A

(k A)
Note: This is different from
(kA) ^{ T } = k A ^{ T }

(A ^{ 1 } ) ^{ T } = (A ^{ T } ) ^{ 1 }

(A + B) ^{ 1 } = A ^{ 1 } + B ^{ 1 }

(AB) ^{ 1 } = B ^{ 1 } A ^{ 1 }